The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. II. Harmonic waves

Abstract

AbstractThe paper presents a comparative analysis of three systems of dynamic equations for fluid‐saturated solids: the exact equations and two simplified versions known as the u‐p approximations obtained by neglecting certain acceleration terms in the exact equations. The constitutive relations for the solid skeleton are written in the general anisotropic incrementally linear form without considering any specific constitutive model or a particular type of anisotropy. The dynamic equations are compared in relation to the existence of solutions in the form of plane harmonic waves. Emphasis is placed on finding conditions for the non‐existence or existence of growing waves whose amplitude increases in time or space as the wave propagates. The conditions are formulated in terms of the acoustic tensor of the skeleton and the compressibility of the pore fluid. In particular, it is shown that for a hyperelastic skeleton, the exact equations and one of the u‐p approximations do not have growing wave solutions, whereas the other u‐p approximation may have such solutions even if the skeleton is hyperelastic.